Summary: A new function in the WRS package compares many quantiles of two distributions simultaneously while controlling the overall alpha error.
When comparing data from two groups, approximately 99.6% of all psychological research compares the central tendency (that is a subjective estimate).
In some cases, however, it would be sensible to compare different parts of the distributions. For example, in reaction time (RT) experiments two groups may only differ in the fast RTs, but not in the long. Measures of central tendency might obscure or miss this pattern, as following example demonstrates.
Imagine RT distributions for two experimental conditions (“black” and “red”). Participants in the red condition have some very fast RTs:
A naïve (but common) approach would be to compare both distributions with a t test:
Results show that both groups do not differ in their central tendency.
Now let’s do the same with a new method!
from the WRS package compares user-defined quantiles of both distributions using a Harrell–Davis estimator in conjunction with a percentile bootstrap. The method seems to improve over other methods: “Currently, when there are tied values, no other method has been found that performs reasonably well. Even with no tied values, method HD can provide a substantial gain in power when q ≤ .25 or q ≥ .75 compared to other techniques that have been proposed”. The method is described in the paper “Comparing two independent groups via the upper and lower quantiles” by Wilcox, Erceg-Hurn, Clark and Carlson (2013).
You can use the function as soon as you install the latest version of the WRS package following this installation instruction.
Let’s compare all percentiles from the 10th to the 90th:
The graphical output shows how groups differ in the requested quantiles, and the confidence intervals for each quantile:
The text output (see below) also shows that groups differ significantly in the 10th, the 50th, and the 60th percentile. The column labeled ‘’.value’’shows the p value for a single quantile bootstrapping test. As we do multiple tests (one for each quantile), the overall Type 1 error (defaulting to .05) is controlled by the Hochberg method. Therefore, for each p value a critical p value is calculated that must be undercut (see column ‘_crit’. The column ‘signify’ marks all tests which fulfill this condition:
To summarize, we see that we have significant differences between both groups: the red group has significantly more faster RTs, but in their central tendency longer RTs.
Recommendations for comparing groups:
Wilcox, R. R., Erceg-Hurn, D. M, Clark, F., & Carlson, M. (in press). Comparing two independent groups via the lower and upper quantiles. Journal of Statistical Computation and Simulation. doi:10.1080/00949655.2012.754026.
This is the evolution of a bivariate correlation between two questionnaire scales, “hope of power” and “fear of losing control”. Both scales were administered in an open online study. The video shows how the correlation evolves from r = .69*** (n=20) to r = .26*** (n=271). It does not stabilize until n = 150.
Data has not been rearranged – it is the random order how participants dropped into the study. This had been a rather extreme case of an unstable correlation – other scales in this study were stable right from the beginning. Maybe this video could help as an anecdotal caveat for a careful interpretation of correlations with small n’s (and with ‘small’ I mean n < 100) …
A little follow up to my testosterone comment (written in German):
When three variables are correlated to each other, and two of the three correlations are known, the range for the third correlation is restricted according to this formula (Olkin, 1981):
As one can see, one, or both of the two given correlations have to be fairly high to imply a positive third correlation.Olkin, I. (1981). Range restrictions for product-moment correlation matrices. Psychometrika, 46, 469-472. doi:10.1007/BF02293804